Balance wheels for watches have previously been principally made of metal. A balance spring is arranged to oscillate the balance wheel and to do so with a period of oscillation that should be isochronous.
The metals used have generally high linear expansion coefficients as compared to most ceramics. This has been the norm for example in the systems containing a balance wheel made of metal (Fe—Ni, Cu—Be, Cu—Zn, Cu—Au, Ni or combinations of these) and the balance spring made from an Fe—Ni alloy, or Fe—Mn—C or other steel derivative alloys. The applicant's earlier application PCT/GB03/003000 filed on 10 Jul. 2003 and published 22 Jan. 2004 and hereby incorporated by reference, discloses new balance spring materials to enable the thermal and magnetic influences in this relationship to be improved or overcome, and thus greater precision to be reached.
The effects of thermal influences upon the balance wheel and the balance spring are not the same. The thermal and thermoelastic characteristics within the relationship between the balance wheel and balance spring do not evolve in an identical manner.
The most successful previous attempts (C. E. Guillaume's bimetallic compensating balance wheel and steel balance spring system invented 1912, Hamilton's precision ferro-nickel based spring alloy in conjunction with steel and invar ovalising balance wheel invented 1943) to bring the terms of the associating relationship into constancy have required the use of materials which despite their useful thermal characteristics (the ferro-nickel alloys have an abnormal Young's modulus evolution) are sensitive to magnetism. This latter influence disturbs the Young's modulus stability and causes negative effects to the precision (isochronism) of the timekeeper.
The expression for the period of oscillation T is described as follows:
                    T        =                  2          ⁢                                          ⁢          π          ⁢                                    I              G                                                          [        1        ]            
T: the period of oscillation, I: the moment of inertia of the balance wheel, G: the torque of the balance spring.
The oscillator system is subject to variations of magnetic and thermal nature. When a balance wheel is made of metal it expands with an increase in temperature. The balance spring that is made most generally of a ferro-nickel alloy also expands with an increase in temperature.
The rate of linear expansion is measured in units of 1/1000 of millimetres/degree Kelvin, represented for example in the case of copper, Cu, as +17×10−6K−1 and is known as the α coefficient.
The thermoelastic coefficient describes the tendency of change in elasticity of the material for a rise in temperature.
The ferro-nickel alloy has a positive thermoelastic coefficient which is described as ‘abnormal’, up to 40° C. when fully de-magnetised. Magnetic accumulation however lowers this threshold which causes the divergence of terms E and r at lower temperature as shown in the graph of FIG. 1 depicting variance of r (radius of gyration of the balance wheel) and √E (square root of the modulus of elasticity of the balance spring) with temperature. This is the cause of the resultant error in the isochronism of the oscillator.
Whereas the balance wheel is in general only affected by thermal variations, which affect its physical dimensions, the balance spring is affected by both thermal and magnetic variations, which affect both its physical dimensions, and its elasticity (Young's modulus).